(b) Show that the charged induced on the hemisphere is: Q = E o a 2 3π (1)

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1 Problem. Defects This problem will study defects in parallel plate capacitors. A parallel plate capacitor has area, A, and separation, D, and is maintained at the potential difference, V = E o D. There are n defects per unit area on the lower plate and none on the upper. The defects consist of hemispherical shells of radius a bending towards the upper plate. You should assume that a D, and that na 2 so that the defects are very widely spaced. (a) Determine the charge per unit area on and near the defect. Plot the surface charge on the hemisphere as a function of θ, and on the plane as a function of r. (Hint: To solve for the potential in the vicinity of a defect use that fact that for a z D the potential reaches its unperturbed form Φ(z) = E o z, so that the upper boundary can be ignored.) (b) Show that the charged induced on the hemisphere is: Q = E o a 2 3π () (c) Use these results to show that the capacitance is unchanged by the defect to the order we are working, i.e. C A d (d) In deriving this result we have used that D a. The size of corrections to the potential you found are of order a 3 /D 3. Explain why. (2)

2 Problem 2. Force between two rings of charge A single ring of charge of radius a and total charge Q is centered at the origin and lies in the xy plane. (a) Show that the potential far from the ring can be written as the multipole expansion Φ = Q 4π l a l P l () r l+ P l (cos θ) (3) Q 4πr Qa 2 2 4πr P 2(cos θ) + 3 Qa πr P 4(cos θ) +... (4) 5 where θ is measured relative to the z axis, and were in the second line we have used the known values for P l (). What are the values of the spherical multipoles q lm (up to l = 2), and the cartesian multipoles p i and Q ij. (b) For a ring of charge of radius a, use an elementary argument to determine the potential along the z axis. Verify that it agrees with the expansion of part (a) when part (a) is evaluated on the z axis. (c) Show that the force between two coaxial charged rings of charge Q and Q widely separated by a distance, 2Z, along the z axis is F where a negative answer indicates an attractive force. Q2 6πZ + 3 Q2 a πZ +... (5) 4 An elegant way to find this is to use the Green Reciprocity theorem which is equivalent to the statement that G D (r, r ) = G D (r, r). In this context, use this condition to show that the potential energy of a quadrupole charge distribution in the electrostatic potential from a monopole is the same as the potential energy of a monopole in an electrostatic potential from a quadrupole. 2

3 Problem 3. A ring of charge close to a plane (a) Consider a long line of charge separated from a grounded plane by separation z o. The charge per length is λ. Determine the force per length between the grounded plane and the charged line. (b) By integrating the force found in part(a), show that the potential energy per length of the line of charge and the grounded plane is u int = λ2 4π log 2z o + const (6) This potential energy for is exactly half of the potential energy between the line of charge and its image. Qualitatively explain why this is the case. (c) Conser a ring of radius a and total charge Q, separated from a plane by a height z o. Use the results of this problem to determine the total force between the ring and the plane when z o a. Explain qualitatively why the results of this problem apply. 3

4 Problem 4. The free green function in cylindrical coordinates (a) Show that the green function in cylindrical coordinates can be expanded as G(r, r o ) = kdk [ e imφ J m (kρ) ] [ e imφo J m (kρ o ) ] g km (z, z o ) (7) and determine the appropriate equation for g km (z, z o ). (Hint: this may be a good time to examine the course notes and to write δ 3 (r r o ) = ρ δ(ρ ρ o)δ(z z o )δ(φ φ o ) as an expansion in eigen functions in the ρ, φ directions) (b) (Optional) If you dont know what a Bessel function looks like, plot J (x), J (x), J 2 (x) and record their series expansions at small and large x. Be aware of the following indentity in 2 dimensions: the 2D function e ik r (8) can be written as a fourier series at each radius r. Defining r = r (cos φ, sin φ), we have e ik r cos φ = e imφ i m J m (kr ) (9) (c) Consider a two dimensional function f(r ) which is independent of the azimuthal angle φ. Its Fourier transform, ˆf(k ) = d 2 x e ik r, is independent of the azimuthal of the k. Using Eq. (9), determine an integral relation between ˆf(k ) and f(r ) (and vice versa) using the Bessel function J (k r ). We say that ˆf(k ) and f(r ) are (up to a constant) Hankel transforms of each other (google Hankel transform). (d) Use the methods discussed in class (Zangwill calls this the method of direct integration) to show that the free Green function in cylindrical coordinates can be written G o (r, r o ) 4π r r o = where z > and z < is the greater and lesser of z and z o. It is useful to compare this result to the one derived in class kdk [ e imφ J m (kρ) ] [ e imφo J m (kρ o ) ] e k(z> z<) 2k () 4π r r o = l= l l [Y lm (θ, φ)y lm(θ o, φ o )] 2l + r< l r> l+ () and to a similar problem that could have been asked 4π r r o = dk [ e im(φ φ o) e ik(z zo)] I m (kρ < )K m (kρ > ) (2) 4

5 Problem 5. The potential energy of a charged ring A charged ring of radius a and total charge Q is at a height z o above a grounded plane. (a) Show that the interaction energy between the plane and the ring is U int (z o ) = U(z o ) U self = d 3 r d 3 r ρ(r) [G(r, r ) G o (r, r )] ρ(r ) (3) 2 ring ring where G(r, r ) is the green function of a point charge in the presence of the grounded plane, and G o (r, r ) is the free green function. (b) From the image solution for the Green function and the expansion given in Eq. (), show that the interaction energy of a ring with a grounded potential is U int (z o ) = Q2 8πa dx [J (x)] 2 e 2x(zo/a) (4) The last remaining integral can be done (with Mathematica) [ ] U int (z o ) = Q2 a 8πa zπ EllipticK( a2 z ) 2 (5) (c) Starting from the integral in Eq. (4) and the expansion of the Bessel function (see DLMF), determine the asymptotic form of the force on the ring for z o a. You should find that your result is in agreement with Eq. (5). (d) When z a, use the series expansions of complete elliptic integrals available in Mathematica (FullSimplify[Series[EllipticK[-y],...], Assumptions->{y>}] worked for me), to show that the potential energy the ring and the plane is: U int (z o ) Q2 8π 2 a log(z o/4a) (6) Compute the force and verify consistency with Eq. (6) and its corresponding force. (e) Use Mathematica or other program to plot the potential energy U int (z o )/(Q 2 /4πa) versus z o /a, together with the short and long distance asymptotics all in one plot. 5